Topological mixing in 𝐶𝐴𝑇(-1)-spaces

Charitos, Charalambos; Tsapogas, Georgios

doi:10.1090/S0002-9947-01-02862-8

Topological mixing in $CAT\left (-1\right )$-spaces
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by Charalambos Charitos and Georgios Tsapogas

Trans. Amer. Math. Soc. 354 (2002), 235-264

DOI: https://doi.org/10.1090/S0002-9947-01-02862-8

Published electronically: August 21, 2001

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Abstract:

If $X$ is a proper $CAT\left ( -1\right )$-space and $\Gamma$ a non-elementary discrete group of isometries acting properly discontinuously on $X,$ it is shown that the geodesic flow on the quotient space $Y=X/\Gamma$ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary $\partial X$ and the non-wandering set of the flow equals the whole quotient space of geodesics $GY:=GX/ \Gamma$ (the latter being redundant when $Y$ is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete $CAT\left ( -1\right )$-spaces by a one-ended group of isometries and (C) finite $n$-dimensional ideal polyhedra.

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